Solving a Yohaku


To solve a yohaku, you must fill in the blank space so that the cells give the sum or the product shown in each row and column. 

Some yohakus have no further restrictions. These puzzles have an infinite number of solutions! For example, in the yohaku on the right, the two blank cells on the top row have to add to give 11.  

This can be done in a variety of ways, but suppose you choose to do 4 and 7. Putting these in will give you this on the right:

Now looking at the left-hand column, we need to think of what must be added to 4 to total 12. At the same time, looking at the right-hand column, what must be added to 7 to give 13? A bit of thought will give the solution as shown.

Most yohakus, though, have a restriction (written below the puzzle) that needs to be satisfied. These will require a bit more thought to solve.

For example, in this multiplicative yohaku (shown right) the restriction states 'Use only whole numbers'. You might look at the top row and start thinking of two numbers that multiply to give 40. Initially you might try 5 and 8. However, now you must think of a number that you multiply 5 by to give 32. You can't do this using whole numbers, so you will have to try a different pair of numbers, say 4 and 10. This does allow you to use whole numbers to complete the problems (as hinted at on the right).

Looking carefully at the sum or product in each row or column will give you some clues as to how these values can be decomposed and, combined with information from other rows and columns, will help you find the values for certain cells. For example, in this 3-by-3 yohaku, the restriction is 'Use prime numbers'. Looking at the top row, you need to think of three prime numbers that multiply to give 8. 

There is only one way to do this i.e. 2×2×2. Putting these in, we can now look at the left-hand column for further clues: what values could we use so that  2 times a prime times a prime gives 50?

Again, there is only one way of doing this as shown below on the right. With a little more thought, it should be possible to fill in the remaining cells.

Some yohakus will have more than one possible solution. Many will involve a fair amount of trial and error.